# General tools to evaluate closed forms of series

• Nov 23rd 2013, 07:38 PM
mathnubsauce
General tools to evaluate closed forms of series
Hello, this is my first post on MHF.

I was wondering if someone might be able to provide a general set of machinery and tools to evaluate various series (supposing they converge) into their closed form. I know this might not be able to be done for every infinite series.
Sum of the tools I know of are:
Geometric series in the finite and the infinite sum case
Expression for (1-x)^(-k)
sines, cosines, exp

For example, if I wanted to evaluate a series such as: (sum from n=1 to infinity) of 1/((3*n-1)*(3*n-2)). I can very well put this into Wolfram and get the desired answer but for my personal use, how would one evaluate such a series?

• Nov 23rd 2013, 10:00 PM
romsek
Re: General tools to evaluate closed forms of series
this looks pretty good at first glance Pauls Online Notes : Calculus II - Series & Sequences
• Nov 23rd 2013, 10:29 PM
JJacquelin
Re: General tools to evaluate closed forms of series
Hi !
a beautifull tool in cases of series of polynomial fractions is the use of polygamma functions.
More generally, many series are particular cases of generalized hypergeometric series. After expressing the series as an hypergeometric function, one look if the parameters corresponds to a special function of lower level, in order to reduce the closed form to a simpler function.
For example, see pages 26-28 in the paper "Safari in the Contry of Special Functions" published on Sribd :
http://www.scribd.com/JJacquelin/documents
• Dec 20th 2013, 05:05 PM
Spacemoss
Re: General tools to evaluate closed forms of series